Showing papers on "Probability-generating function published in 2012"

Criticality and self-organization in branching processes: application to natural hazards

Abstract: The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that the expected value of the energy does not exist (is infinite), and in practice, that the mean of a finite set of data in not representative of the full population. Also, the distribution presents scale invariance, which implies that it is not possible to define a characteristic scale for the energy. A simple model to account for this peculiar statistics is a branching process: the activation or slip of a fault segment can trigger other segments to slip, with a certain probability, and so on. Although not recognized initially by seismologists, this is a particular case of the stochastic process studied by Galton and Watson one hundred years in advance, in order to model the extinction of (prominent) families. Using the formalism of probability generating functions we will be able to derive, in an accessible way, the main properties of these models. Remarkably, a power-law distribution of energies is only recovered in a very special case, when the branching process is at the onset of attenuation and intensification, i.e., at criticality. In order to account for this fact, we introduce the self-organized critical models, in which, by means of some feedback mechanism, the critical state becomes an attractor in the evolution of such systems. Analogies with statistical physics are drawn. The bulk of the material presented here is self-contained, as only elementary probability and mathematics are needed to start to read.

A retrial system with two input streams and two orbit queues

Abstract: Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented.

Prediction of molecular weight distributions in polymers using probability generating functions

Abstract: We review the work of our group in the area of mathematical modelling of polymerisation reactors and processes, with emphasis in the prediction of molecular weight distributions using probability generating functions (pgfs). We apply the method to a polymerisation example for which the analytic solution is known, and show that its predictions are very accurate. We also compare the performance of the method to that of the straightforward integration of the mass balances of the reacting system, showing that the pgf method results in very fast calculations. We end by discussing ongoing and future applications of the method. © 2011 Canadian Society for Chemical Engineering

Analysis of a discrete-time queue with geometrically distributed service capacities

Abstract: We consider a discrete-time queueing model whereby the service capacity of the system, ie, the number of work units that the system can perform per time slot, is variable from slot to slot Specifically, we study the case where service capacities are independent from slot to slot and geometrically distributed New customers enter the system according to a general independent arrival process Service demands of the customers are iid and arbitrarily distributed For this (non-classical) queueing model, we obtain explicit expressions for the probability generating functions (pgf's) of the unfinished work in the system and the queueing delay of an arbitrary customer In case of geometric service demands, we also obtain the pgf of the number of customers in the system explicitly By means of some numerical examples, we discuss the impact of the service process of the customers on the system behavior

Limit theorems for the Green function of the lattice Laplacian under large deviations of the random walk

Abstract: We carry out a resolvent analysis of the lattice Laplacian (the generator of a simple random walk on the -dimensional integer lattice) under large deviations of the random walk. This enables us to obtain asymptotic representations for the transition probability of the simple random walk and the corresponding Green function. We explicitly describe the asymptotic behaviour of the transition probability as the spatial and temporal variables jointly tend to infinity. The resulting Cramer-type expansion for the transition probability is 'universal' in this sense. In particular, it enables us to construct a scale for measuring the transition probability as a function of the time assuming that the spatial variable is of order for various values of . We prove limit theorems on the asymptotic behaviour of the Green function of the transition probabilities under large deviations of the random walk.

On a bilateral birth-death process with alternating rates

Abstract: We consider a bilateral birth-death process characterized by a constant transition rate λ from even states and a possibly different transition rate μ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.

Moment identity for discrete random variable and its applications

Abstract: In this paper, we obtain a moment identity applicable to a general class of discrete probability distributions. We then derive the corresponding identities for modified power series, Ord and Katz families. It is noted that the proposed identity has potential applications in different fields.

Analysis of a two-class FCFS queueing system with interclass correlation

Abstract: This paper considers a discrete-time queueing system with one server and two classes of customers. All arriving customers are accommodated in one queue, and are served in a First-Come-First-Served order, regardless of their classes. The total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. The classes of consecutively arriving customers, however, are correlated in a Markovian way, i.e., the probability that a customer belongs to a class depends on the class of the previously arrived customer. Service-time distributions are assumed to be general but class-dependent. We use probability generating functions to study the system analytically. The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the (average) number of customers in the system and the (average) customer delay and customer waiting time.

Choice probability generating functions

Abstract: This paper considers discrete choice, with choice probabilities coming from maximization of preferences from a random utility field perturbed by additive location shifters (ARUM). Any ARUM can be characterized by a choice-probability generating function (CPGF) whose gradient gives the choice probabilities, and every CPGF is consistent with an ARUM. We relate CPGF to multivariate extreme value distributions, and review and extend methods for constructing CPGF for applications.

Combined Analysis of Transient Delay Characteristics and Delay Autocorrelation Function in the G e o X / G /1 Queue

Abstract: We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state.

Investigation of Probability Generating Function in an Interdependent M/M/1/:(; GD) Queueing Model with Controllable Arrival Rates Using Rouche's Theorem

Abstract: Present paper deals a M/M/1:(∞; GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration; which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

Branching processes, criticality, and self-organization: application to natural hazards

Abstract: The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that the expected value of the energy does not exist (is infinite), and in practice, that the mean of a finite set of data in not representative of the full population. Also, the distribution presents scale invariance, which implies that it is not possible to define a characteristic scale for the energy. A simple model to account for this strange statistics is a branching process: the activation or slip of a fault segment can trigger other segments to slip, with a certain probability, and so on. Although not recognized initially by seismologists, this is a particular case of the stochastic process studied by Galton and Watson one hundred years in advance, in order to model the extinction of (prominent) families. Using the formalism of probability generating functions we will be able to derive, in an accessible way, the main properties of these models. Remarkably, a power-law distribution of energies is only recovered in a very special case, when the branching process is at the onset of attenuation and intensification, i.e., at criticality. In order to account for this fact, we introduce the self-organized critical models, in which, by means of some feedback mechanism, the critical state becomes an attractor in the evolution of such systems. Analogies with statistical physics are drawn. The bulk of the material presented here is self-contained, as only elementary probability and mathematics, as that that should be taught in high school, is needed to start to read.

Unified uncertainty analysis by the extension universal generating functions

Abstract: In this paper, a unified uncertainty analysis method based on the extension universal generating function is proposed for engineering problems described by the mixture of random, interval and p-box variables. In the method, the univariate approximation approach is extended for mixed variables, and then the performance function is divided into three parts. Traditional universal generating function can only model the case that all variables exist in system are random variables. However, in order to calculate the bounds of system probability of failure under mixed variables, the mixed universal generating function is developed to extend the traditional universal generating function. The optimization models based on the mixed universal generating function are presented to calculate system probability of failure under mixed variables. An engineering example is used to validate the proposed method.

Distribution and double generating function of number of patterns in a sequence of Markov dependent multistate trials

Abstract: In this manuscript, the dual relationship between the probability of number of runs and patterns and the probability of waiting time of runs and patterns in a sequence of multistate trials has been studied via double generating functions and recursive equations. The results, which are established under different assumptions on patterns, underlying sequences and counting schemes, are extensions of Koutras’s results (1997, Advances in Combinatorial Methods and Applications to Probability and Statistics, Boston: Birkhauser). As byproducts, the exact distributions of the longest-run statistics are also derived. Numerical examples are provided for illustrating the theoretical results.

The dimension reduction algorithm for the positive realization of discrete phase-type distributions

Abstract: This paper provides an efficient dimension reduction algorithm of the positive realization of discrete phase type(DPH) distributions. The relationship between the representation of DPH distributions and the positive realization of the positive system is explained. The dimension of the positive realization of a discrete phase-type realization may be larger than its McMillan degree of probability generating functions. The positive realization with sufficient large dimension bound can be obtained easily but generally, the minimal positive realization problem is not solved yet. We propose an efficient dimension reduction algorithm to make the positive realization with tighter upper bound from a given probability generating functions in terms of convex cone problem and linear programming.

A Single Server Compulsory Vacation Queue with Two Type of Services and with Restricted Admissibility

Abstract: A single server queue with compulsory vacation has been considered. In addition admission to queue is based on a Bernoulli process and the server gives two types of services. For this model the probability generating functions of number of customers in the queue for different server states are obtained using supplementary variable technique. Some performance measures are calculated. Particular cases are deduced and some numerical examples are presented.

On total progeny of multitype Galton-Watson process and the first passage time of random walk with bounded jumps

Abstract: In this paper, we first form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the hitting time of corresponding random walks.

Geometric Series via Probability

Abstract: We describe a probability theory approach to find the sum of a convergent geometric series.

Likelihood Ratio and Strong Limit Theorems for the Discrete Random Variable

Abstract: This in virtue of the notion of likelihood ratio and the tool of moment generating function, the limit properties of the sequences of random discrete random variables are studied, and a class of strong deviation theorems which represented by inequalities between random variables and their expectation are obtained. As a result, we obtain some strong deviation theorems for Poisson distribution and binomial distribution.

Bounds for the probability generating functional of a Gibbs point process

Abstract: We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics like the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity and higher order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

On the limiting distribution of the maximum with random sample size of regular variation function

Abstract: Let X 1 , X 2 , … be i.i.d. sequence and ƒ(x) be a regular variation function with index r. Let N(n) be a positive integer valued random variable and equation in probability, where η is a positive random variable. When some conditions are satisfied, the asymptotic distribution of M ƒ(N(n)) is derived.

Important Functions and Their Features

Abstract: Probability theory can be understood as a particular field in mathematics. Hence, it is only to be expected that it relies intensely on theory from analysis and algebra. For example, the fact that the cumulative probability over all values a random variable can assume has to be equal to one is not always feasible to check for without a profound knowledge of mathematics. Continuous probability distributions involve a good deal of analysis and the more sophisticated a distribution is, the more mathematics is necessary to handle it. Keywords: monotonic functions; continuous function; continuous; derivative; monotonically increasing; strictly monotonic increasing; A continuous function ; monotonically decreasing; strictly monotonic increasing; integral; integral; factorial; gamma function; beta function; Bessel function; NIG; characteristic function

Choice Probability Generating Functions

Abstract: This paper considers discrete choice, with choice probabilities coming from maximization of preferences from a random utility field perturbed by additive location shifters (ARUM). Any ARUM can be characterized by a choice-probability generating function (CPGF) whose gradient gives the choice probabilities, and every CPGF is consistent with an ARUM. We relate CPGF to multivariate extreme value distributions, and review and extend methods for constructing CPGF for applications.

Bridging a Gap in Independent Component Analysis

Abstract: Independent Component Analysis (ICA) is a powerful method which aims at representing a given random signal as a sum of independent sources. The engineering community, however, works at the distribution function or characteristic function level while makes assertions at the random variable level. This legitimacy of this jump has never been established and consists of a longstanding gap in the ICA literature. In this paper, it is proved that existence of a factorization of characteristic function does imply existence of a corresponding decomposition of random variable into independent sum and thus the gap is bridged. The proof relies on two nontrivial results from probability theory.